Nuprl Lemma : pscm-ap-type_wf
∀[C:SmallCategory]. ∀[Gamma,Delta:ps_context{j:l}(C)]. ∀[A:{Gamma ⊢ _}]. ∀[s:psc_map{j:l}(C; Delta; Gamma)].
((A)s ∈ Delta ⊢ )
Proof
Definitions occuring in Statement :
pscm-ap-type: (AF)s,
presheaf-type: {X ⊢ _},
psc_map: A ⟶ B,
ps_context: __⊢,
uall: ∀[x:A]. B[x],
member: t ∈ T,
small-category: SmallCategory
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
presheaf-type: {X ⊢ _},
pscm-ap-type: (AF)s,
and: P ∧ Q,
subtype_rel: A ⊆r B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
all: ∀x:A. B[x],
cat-comp: cat-comp(C),
compose: f o g,
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
prop: ℙ
Lemmas referenced :
pscm-ap_wf,
ps_context_cumulativity2,
small-category-cumulativity-2,
subtype_rel_self,
psc_map_wf,
I_set_wf,
subtype_rel-equal,
psc-restriction_wf,
equal_wf,
pscm-ap-restriction,
iff_weakening_equal,
cat-arrow_wf,
cat-comp_wf,
squash_wf,
true_wf,
psc-restriction-comp,
cat-id_wf,
psc-restriction-id,
presheaf-type_wf,
ps_context_wf,
small-category_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
promote_hyp,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
instantiate,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
universeIsType,
independent_isectElimination,
imageElimination,
universeEquality,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionIsType,
lambdaFormation_alt,
independent_pairFormation,
productIsType,
equalityIstype,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[Gamma,Delta:ps\_context\{j:l\}(C)]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[s:psc\_map\{j:l\}(C;
Delta;
Gamma)].
((A)s \mmember{} Delta \mvdash{} )
Date html generated:
2020_05_20-PM-01_26_12
Last ObjectModification:
2020_04_01-AM-11_00_55
Theory : presheaf!models!of!type!theory
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